The generation of entangled photons is an essential element in current approaches to quantum information processing and quantum communication. In particular, it is often desirable to generate entangled photons having narrow spectral bandwidths to facilitate coupling to long-lived quantum memories or repeaters.
Various physical mechanisms have been considered for generating entangled photons. One such mechanism is four-wave mixing (FWM). In particular, FWM in chip-scale microresonators has been proposed as potentially advantageous means for producing narrowband entangled photons.
FWM in microresonators is usually performed in systems, such as disk and ring resonators, that have azimuthal symmetry (also referred to herein as “cylindrical” symmetry). Such systems are advantageous because, among other reasons, they readily permit near-equal mode spacing, small mode volumes, and high optical quality factors.
For entangled photon production, an underlying electronic nonlinearity is utilized to produce spontaneous degenerate FWM. (Driven FWM has also been considered. In particular, it has been shown that driving above the parametric oscillation threshold can produce frequency combs.) More specifically, if the resonant disk or ring is composed of a material with a third-order nonlinear suceptiblity χ(3) (ρ, z), then upon pumping the resonator with a strong pump beam of frequency ωp selected to couple into mode m of the ring, a process may occur by spontaneous FWM in which two pump photons are annihilated to produce an entangled pair of signal and idler photons of respective frequencies ωs and ωi. However, the process must conserve both energy and momentum. The conservation of energy requires that ωs+ωi=2ωp.
The conservation of momentum requires that ms+mi=2mp, in which ms, mi, and mp are the respective mode numbers for the signal, idler, and pump waves.
Various investigators have attempted to theoretically model the spectrum resulting from spontaneous and driven FWM in azimuthally symmetric systems. Although it has been more common in these efforts to approximate the cylindrical resonator as an unfolded straight waveguide, several investigators have employed a representation in cylindrical coordinates, which is more naturally suited to systems with azimuthal symmetry.
For example, a fully vectorial treatment of cascaded FWM in a spherical resonator, taking into account all components of the electric field, spatial mode overlaps, and resonator dispersion profiles was reported in Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010) and in Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010). Although those results apply only to spherical resonators and neglect material dispersion, they successfully track the spectral evolution resulting from the FWM interaction.
A theoretical study of the quantum correlations resulting from spontaneous FWM in azimuthally symmetric ring resonators was presented in J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470-1483 (2011). That work predicted that FWM in an azimuthally symmetric ring resonator could be used to generate a frequency-bin entangled comb of photon pairs. However, that work included certain implicit assumptions based on rectilinear rather than cylindrical resonator geometries.
More specifically, in their quantum formalism the authors enforced a phase-matching condition that involved integrating an effective linear group velocity and linear wavevector (i.e., k-vector) over an unfolded cavity of specified linear length. In a cylindrical system, however, the linear momentum is zero and the linear wavevector and group velocities are poorly defined, particularly for small resonator radii, such as radii approaching ten or fewer times the resonant wavelength. Moreover, because cylindrical resonator systems lack translational symmetry, the accuracy with which optical wave propagation can be modeled using a linear wavevector is limited. Greater accuracy would be achieved using the angular wavevector and angular group velocities.
One thing that has been lacking, until now, is a fully vectorial quantum model in cylindrical coordinates to describe spontaneous FWM in axially symmetric systems, which might facilitate new design approaches and even lead to new optical devices based on properties revealed by the use of such a model.